Questions — Edexcel (9685 questions)

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Edexcel FP2 2018 June Q5
9 marks Challenging +1.2
5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 0$$ Given that at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
  1. show that, at \(x = 0 , \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 3 } { 2 }\)
  2. Find a series solution for \(y\) up to and including the term in \(x ^ { 3 }\)
Edexcel FP2 2018 June Q6
13 marks Standard +0.8
6. (a) Find the general solution of the differential equation $$6 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 6 y = x - 6 x ^ { 2 }$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 }\) when \(x = 0\)
Edexcel FP2 2018 June Q7
12 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-24_508_896_212_525} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 2 + \sqrt { 3 } \cos \theta , \quad 0 \leqslant \theta < 2 \pi$$ The tangent to \(C\) at the point \(P\) is parallel to the initial line.
  1. Show that \(O P = \frac { 1 } { 2 } ( 3 + \sqrt { 7 } )\)
  2. Find the exact area enclosed by the curve \(C\).
Edexcel FP2 2018 June Q8
13 marks Standard +0.8
8. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find $$\int 2 x ^ { 5 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ (b) Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). Given that \(y = 0\) when \(x = 1\) (c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}]{5aa7f449-215b-4a21-9fdc-df55d26abc9d-32_2632_1826_121_121}
Edexcel FP2 Q1
6 marks Moderate -0.8
  1. (a) Sketch, on the same axes, the graph with equation \(y = | 3 x - 1 |\), and the line with equation \(y = 4 x + 3\).
Show the coordinates of the points at which the graphs meet the \(x\)-axis.
(b) Solve the inequality \(| 3 x - 1 | < 4 x + 3\).
Edexcel FP2 Q2
5 marks Standard +0.3
2. (a) Express \(\frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions.
(b) Hence prove that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 n } { 3 ( 2 n + 3 ) }\).
Edexcel FP2 Q3
7 marks Standard +0.3
3. (a) Given that \(y = \ln ( 1 + 5 x ) , | x | < 0.2\), find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(b) Hence obtain the M aclaurin series for \(\ln ( 1 + 5 x ) , | x | < 0.2\), up to and including the term in \(x ^ { 3 }\).
Edexcel FP2 Q4
8 marks Challenging +1.2
4. Use the Taylor Series method to find the series solution, ascending up to and including the term in \(x ^ { 3 }\), of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } = 3 x + 4$$ given that \(\frac { \mathrm { dy } } { \mathrm { dx } } = y = 1\) at \(x = 0\).
(Total 8 marks)
Edexcel FP2 Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{affb668f-4d43-4fa8-a5b7-d536a58126b9-3_529_668_223_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has polar equation, \(r = 2 + \sin 3 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) Use integration to calculate the exact value of the area enclosed by \(C\), the line \(\theta = 0\) and the line \(\theta = \frac { \pi } { 2 }\).
Edexcel FP2 Q6
9 marks Challenging +1.2
6. (a) Use de M oivre's Theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta .$$ (b) Hence or otherwise, prove that the only real solutions of the equation $$\sin 5 \theta = 5 \sin \theta ,$$ are given by \(\theta = n \tau\), where \(n\) is an integer.
Edexcel FP2 Q7
9 marks Standard +0.3
7. A population \(P\) is growing at a rate which is modelled by the differential equation $$\frac { d P } { d t } - 0.1 P = 0.05 t$$ where \(t\) years is the time that has elapsed from the start of observations.
It is given that the population is 10000 at the start of the observations.
  1. Solve the differential equation to obtain an expression for \(P\) in terms of \(t\).
  2. Show that the population doubles between the sixth and seventh year after the observations began.
    (2)
Edexcel FP2 Q8
11 marks Standard +0.8
8. A complex number \(z\) satisfies the equation $$| z - 5 - 12 i | = 3$$
  1. Describe in geometrical terms with the aid of a sketch, the locus of the point which represents \(z\) in the A rgand diagram. For points on this locus, find
  2. the maximum and minimum values for \(| z |\),
  3. the maximum and minimum values for arg \(z\), giving your answers in radians to 2 decimal places.
Edexcel FP2 Q9
13 marks Challenging +1.2
9. Resonance in an electrical circuit is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } } + 64 V = \cos 8 t$$ where \(V\) represents the voltage in the circuit and \(t\) represents time.
  1. Find the value of \(\lambda\) for which \(\lambda\) tsin8t is a particular integral of the differential equation.
  2. Find the general solution of the differential equation. Given that \(V = 0\) and \(\frac { \mathrm { d } V } { \mathrm {~d} t } = 0\) when \(t = 0\),
  3. find the particular solution of the equation.
  4. Describe the behaviour of \(V\) as \(t\) becomes large, according to this model.
Edexcel FP2 Specimen Q1
7 marks Standard +0.3
  1. Find the set of values of \(x\) for which
$$\frac { x } { x - 3 } > \frac { 1 } { x - 2 }$$
Edexcel FP2 Specimen Q2
5 marks Standard +0.8
  1. (a) Express as a simplified single fraction \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } }\) (b) Hence prove, by the method of differences, that
$$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
Edexcel FP2 Specimen Q3
6 marks Challenging +1.2
  1. (a) Show that the transformation \(T\)
$$w = \frac { z - 1 } { z + 1 }$$ maps the circle \(| z | = 1\) in the \(z\)-plane to the line \(| w - 1 | = | w + \mathrm { i } |\) in the \(w\)-plane. The transformation \(T\) maps the region \(| z | \leq 1\) in the \(z\)-plane to the region \(R\) in the \(w\)-plane.
(b) Shade the region \(R\) on an Argand diagram.
Edexcel FP2 Specimen Q4
7 marks Challenging +1.8
4. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } = x , \quad y = 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \text { at } x = 1$$ Find a series solution of the differential equation in ascending powers of ( \(x - 1\) ) up to and including the term in \(( x - 1 ) ^ { 3 }\).
Edexcel FP2 Specimen Q5
10 marks Standard +0.3
5. (a) Obtain the general solution of the differential equation $$\frac { \mathrm { d } S } { \mathrm {~d} t } - 0.1 S = t$$ (b) The differential equation in part (a) is used to model the assets, \(\pounds S\) million, of a bank \(t\) years after it was set up. Given that the initial assets of the bank were \(\pounds 200\) million, use your answer to part (a) to estimate, to the nearest \(\pounds\) million, the assets of the bank 10 years after it was set up.
Edexcel FP2 Specimen Q6
12 marks Challenging +1.2
6. The curve \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta , \quad \frac { - \pi } { 4 } \leq \theta \leq \frac { \pi } { 4 }$$
  1. Sketch the curve \(C\).
  2. Find the polar coordinates of the points where tangents to \(C\) are parallel to the initial line.
  3. Find the area of the region bounded by \(C\).
Edexcel FP2 Specimen Q7
14 marks Standard +0.8
7. (a) Given that \(x = e ^ { t }\), show that
  1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \frac { \mathrm {~d} y } { \mathrm {~d} t }$$
  2. $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$ (b) Use you answers to part (a) to show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = \mathrm { e } ^ { 3 t }$$ (c) Hence find the general solution of $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$
Edexcel FP2 Specimen Q8
14 marks Challenging +1.2
  1. (a) Given that \(z = e ^ { i \theta }\), show that
$$z ^ { p } + \frac { 1 } { z ^ { p } } = 2 \cos p \theta$$ where \(p\) is a positive integer.
(b) Given that $$\cos ^ { 4 } \theta = A \cos 4 \theta + B \cos 2 \theta + C$$ find the values of the constants \(A , B\) and \(C\). The region \(R\) bounded by the curve with equation \(y = \cos ^ { 2 } x , - \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }\), and the \(x\)-axis is rotated through \(2 \pi\) about the \(x\)-axis.
(c) Find the volume of the solid generated.
Edexcel F3 2021 January Q1
6 marks Challenging +1.2
  1. Relative to a fixed origin \(O\), the points \(A\), \(B\), \(C\) and \(D\) have coordinates \(( 0,4,1 ) , ( 4,0,0 )\), \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
    1. Determine the exact area of triangle \(A B C\).
    2. Determine in terms of \(k\), the volume of the tetrahedron \(A B C D\), simplifying your answer. \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
    3. Determine the exact area of triangle \(A B C\).
    $$\text { etrahedron } A B C D \text {, simplifying }$$
Edexcel F3 2021 January Q2
6 marks Standard +0.8
2. $$y = \ln ( \tanh 2 x ) \quad x > 0$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosech } 4 x$$ where \(p\) is a constant to be determined.
  2. Hence determine, in simplest form, the exact value of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
Edexcel F3 2021 January Q3
6 marks Standard +0.3
3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & k & 2 \\ 2 & 2 & k \\ 1 & 2 & 2 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Determine the values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
  2. find \(\mathbf { A } ^ { - 1 }\), giving your answer in terms of \(k\).
    3.
Edexcel F3 2021 January Q4
6 marks Challenging +1.2
4. Using the substitution \(x = 4 \cosh \theta\) show that $$\int \frac { 1 } { \left( x ^ { 2 } - 16 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { a x } { \sqrt { x ^ { 2 } - 16 } } + c \quad | x | > 4$$ where \(a\) is a constant to be determined and \(c\) is an arbitrary constant.
(6)